Integral trace forms associated to cubic extensions

Mantilla-Soler, Guillermo

doi:10.2140/ant.2010.4.681

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Integral trace forms associated to cubic extensions

Guillermo Mantilla-Soler

Vol. 4 (2010), No. 6, 681–699

DOI: 10.2140/ant.2010.4.681

Abstract

Given a nonzero integer $d$ , we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant $d$ . However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form $q_{K} : {tr}_{K ∕ ℚ} (x^{2}) |_{O_{K}^{0}}$ as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element $T_{K}$ in Bhargava’s class group $Cl (ℤ^{2} \otimes ℤ^{2} \otimes ℤ^{2}; - 3 d)$ such that $q_{K}$ is completely determined by $T_{K}$ . By using one of Bhargava’s composition laws, we show that $q_{K}$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant.

Keywords

integral trace forms, cubic fields, Bhargava's class group, discriminants of number fields

Mathematical Subject Classification 2000

Primary: 11E12

Secondary: 11R29, 11R16, 11E76

Milestones

Received: 18 June 2009

Revised: 5 December 2009

Accepted: 15 May 2010

Published: 25 September 2010

Authors

Guillermo Mantilla-Soler
Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53705 United States

Vol. 4, No. 6, 2010